Finite Group Databasebeta

← #29577865

Subgroups — #29577865

Order 15616.

542 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454900C211yesmaximal, characteristic, complemented
2#21450724C211yesmaximal, characteristic
3#21444243C212yesmaximal
4#21443679C212yesmaximal, complemented
5C61×#1801C211yesmaximal, characteristic, complemented
6C2×#14519753—21no—
7C2×C4×D244C2211yescharacteristic
8Q8×#92418—24no—
9#16839815—22no—
10#16839815C2212yes—
11#16839692C412yes—
12#16839679C412yes—
13#16839653—22no—
14#16839628C2211yescharacteristic
15#16839619—21no—
16#16839489—24no—
17C2×C4×Q8×C61C2211yescharacteristic
18C61×#423C2212yes—
19C2×#101×C61—21no—
20C61×#343—21no—
21C22×D244C2×C411yescharacteristic
22C22×C4×D61—21no—
23#14519753—42no—
24#14519753—42no—
25C4×D244—22no—
26C4×D244—22no—
27C2×C4⋊Dic61—21no—
28C2×C4⋊Dic61C2×C411yescharacteristic
29C2×#11758196—22no—
30C2×#11758198—22no—
31C2×#11758198—22no—
32C2×#11758198—21no—
33C4⋊#92418—24no—
34C4⋊#92418—22no—
35C22⋊#92418—22no—
36C4×#92418—24no—
37C4×#92418D414yes—
38#14519684—22no—
39#14519684—21no—
40#14519684C2×C412yes—
41C22×Q8×C61D411yescharacteristic
42C2×M16×C61—22no—
43C4×Q8×C61—22no—
44C4×Q8×C61—22no—
45C2×C4⋊C4×C61C2×C412yes—
46C2×C4⋊C4×C61D411yescharacteristic
47C2×C42×C61D412yes—
48C2×C42×C61C2311yescharacteristic
49C4⋊C8×C61—22no—
50C4⋊C8×C61—22no—
51#101×C61—42no—
52C23×D61—21no—
53C2×Q8×C61—22no—
54C2×Q8×C61—22no—
55C2×Q8×C61D812yes—
56C2×Q8×C61SD1612yes—
57C22×C4×C61C22⋊C412yes—
58C22×C4×C61C22⋊C412yes—
59C22×C4×C61C22×C411yescharacteristic
60C22×C4×C61C2×D411yescharacteristic
61C22×C4×C61C2×D411yescharacteristic
62C22×Dic61—21no—
63C22×Dic61—22no—
64C2×D244—22no—
65C2×D244—42no—
66C2×C4×D61—42no—
67C2×C4×D61—41no—
68C2×C4×D61—41no—
69C2×C4×D61—42no—
70M16×C61—42no—
71M16×C61—42no—
72C2×C8×C61—42no—
73C4⋊C4×C61—24no—
74C4⋊C4×C61—22no—
75C42×C61—22no—
76C42×C61—22no—
77C42×C61C2×D412yes—
78C42×C61#4112yes—
79#11758198—44no—
80#11758198—44no—
81#11758198—41no—
82#11758198—41no—
83C4⋊Dic61—22no—
84C4⋊Dic61—41no—
85C4⋊Dic61—41no—
86#11758196—44no—
87C2×#92418—24no—
88C23×C61C2×C22⋊C411yescharacteristic
89C22×D61—22no—
90C22×D61—41no—
91C22×D61—41no—
92C22×D61—41no—
93C22×D61—41no—
94C22×D61—41no—
95C22×D61—41no—
96Q8×C61—24no—
97Q8×C61—24no—
98C2×C4×C61—22no—
99C2×C4×C61—22no—
100C2×C4×C61—22no—
101C2×C4×C61—41no—
102C2×C4×C61—41no—
103C2×C4×C61—42no—
104C2×C4×C61#10214yes—
105C2×C4×C61C4×D412yes—
106C2×C4×C61#12011yescharacteristic
107C2×C4×C61#12112yes—
108C2×C4×C61#12311yescharacteristic
109C2×C4×C61C2×D811yescharacteristic
110C2×C4×C61C2×SD1611yescharacteristic
111C2×C4×C61#13512yes—
112C2×Dic61—42no—
113C2×Dic61—42no—
114C2×Dic61—41no—
115C2×Dic61—41no—
116C2×Dic61—42no—
117C2×Dic61—41no—
118D244—24no—
119D244—42no—
120C4×D61—82no—
121C4×D61—82no—
122C8×C61—82no—
123#92418—44no—
124#9486—611nomaximal, Sylow
125C22×C61—21no—
126C22×C61—21no—
127C22×C61#38611yescharacteristic
128C22×C61C2×#10211yescharacteristic
129C22×C61#41611yescharacteristic
130C2×D61—42no—
131C2×D61—42no—
132C2×D61—42no—
133C2×D61—81no—
134C2×D61—81no—
135C2×D61—81no—
136C2×D61—81no—
137C4×C61—22no—
138C4×C61—24no—
139C4×C61—21no—
140C4×C61—21no—
141C4×C61—42no—
142C4×C61#46512yes—
143C4×C61#46612yes—
144Dic61—82no—
145Dic61—81no—
146#2982—611no—
147#2536—611no—
148#1801D6111yescharacteristic, complemented
149#1486—612no—
150#1286—612no—
151C2×C61—21no—
152C2×C61—21no—
153C2×C61#189111yescharacteristic
154C2×C61#189311yescharacteristic
155C2×C61#191211yescharacteristic
156D61—82no—
157C2×C22⋊Q8—1221no—
158C2×C4×Q8C2×D6111yescharacteristic
159C2×C4×D4—611no—
160C8×Q8—1224no—
161#431—612no—
162#431—1222no—
163#423C2×D6112yes—
164C2×#101—21no—
165#398—612no—
166#394—612no—
167#388—1222no—
168#382—611no—
169#380—1221no—
170#343—21no—
171D4⋊C8—1224no—
172C61#948611yescharacteristic, complemented, Sylow
173C22×Q8C22⋊D6111yescharacteristic
174C22×D4—611no—
175C23×C4—1221no—
176C2×M16—22no—
177C22⋊Q8—2442no—
178C22⋊Q8—2442no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×D4—1222no—
182C4×D4—1222no—
183C2×C4⋊C4—611no—
184C2×C4⋊C4—1222no—
185C2×C4⋊C4—1221no—
186C2×C4⋊C4C4×D6112yes—
187C2×C4⋊C4C22⋊D6111yescharacteristic
188C2×C22⋊C4—1222no—
189C2×C22⋊C4—1222no—
190C2×C22⋊C4—1221no—
191C2×C42D24412yes—
192C2×C42C22×D6111yescharacteristic
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—1224no—
196C4⋊C8—1222no—
197#101—42no—
198C22⋊C8—1222no—
199C4×C8—614no—
200C4×C8—1224no—
201#95—612no—
202#95—1222no—
203#95—1221no—
204C24—1221no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8#1175819912yes—
208C2×Q8#1175820112yes—
209C2×D4—1222no—
210C2×D4—2442no—
211C22×C4—1222no—
212C22×C4—1221no—
213C22×C4—2442no—
214C22×C4—2441no—
215C22×C4—2441no—
216C22×C4—2442no—
217C22×C4#1175819812yes—
218C22×C4#1175819812yes—
219C22×C4C2×C4×D6111yescharacteristic
220C22×C4C2×D24411yescharacteristic
221C22×C4C2×C22⋊D6111yescharacteristic
222M16—42no—
223M16—42no—
224C2×C8—42no—
225C2×C8—1224no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—1222no—
229C4⋊C4—2444no—
230C4⋊C4—2441no—
231C4⋊C4—2441no—
232C22⋊C4—2444no—
233C22⋊C4—2444no—
234C22⋊C4—2441no—
235C22⋊C4—2441no—
236C42—22no—
237C42—22no—
238C42D4×D6112yes—
239C42#1175822412yes—
240C23—1222no—
241C23—2441no—
242C23—2441no—
243C23—2441no—
244C23—2441no—
245C23—2441no—
246C23—2441no—
247C23C2×#1175819811yescharacteristic
248Q8—24no—
249Q8—24no—
250D4—1224no—
251D4—2442no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—4882no—
259C2×C4—4882no—
260C2×C4—2442no—
261C2×C4—2442no—
262C2×C4—2442no—
263C2×C4—2441no—
264C2×C4—2441no—
265C2×C4—2441no—
266C2×C4#1451969714yes—
267C2×C4#1451973712yes—
268C2×C4#1451974711yescharacteristic
269C2×C4#1451974912yes—
270C2×C4#1451976111yescharacteristic
271C2×C4C2×#1175819911yescharacteristic
272C2×C4C2×#1175820111yescharacteristic
273C2×C4#1451979712yes—
274C8—82no—
275C8—2444no—
276C22—21no—
277C22—21no—
278C22—4881no—
279C22—4881no—
280C22—4881no—
281C22—4881no—
282C22—2442no—
283C22—2442no—
284C22—2442no—
285C22#1683963911yescharacteristic
286C22C2×#1451969711yescharacteristic
287C22#1683975611yescharacteristic
288C4—22no—
289C4—24no—
290C4—21no—
291C4—21no—
292C4—42no—
293C4—4882no—
294C4—4881no—
295C4#1683999312yes—
296C4#1683999612yes—
297C2—21no—
298C2—21no—
299C2—4882no—
300C2#2144623511yescharacteristic
301C2#2144624511yescharacteristic
302C2#2144635411yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1801C2
socleC22×C61#386
prime radicalC22×D244C2×C4
cosocleC2×C42×C61C23
hypercenter#1801D61
hypocenterC61#9486
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21454900C2semidirect
4. #21443679C2semidirect
5. C61×#1801C2semidirect
148. #1801D61semidirect
172. C61#9486semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214549002
#214507242
#214442432
#214436792
C61×#18012
#948661